- Email: [email protected]
THE EFFECT OF REFRACTIVE INDEX ON SIZE DISTRIBUTIONS AND LIGHT SCATTERING COEFFICIENTS DERIVED FROM OPTICAL PARTICLE COUNTERS
PII: S0021-8502(99)00573-X
J. Aerosol Sci. Vol. 31, No. 8, pp. 945}957, 2000 Published by Elsevier Science Ltd Printed in Great Britain 0021-8502/00/$ - see front matter
THE EFFECT OF REFRACTIVE INDEX ON SIZE DISTRIBUTIONS AND LIGHT SCATTERING COEFFICIENTS DERIVED FROM OPTICAL PARTICLE COUNTERS* Yangang LiuR and Peter H. Daum Brookhaven National Laboratory, Environmental Chemistry Division, Bldg. 815E, P.O. Box 5000, Upton, NY 11973-5000, USA (First received 7 June 1999; and in ,nal form 18 November 1999) Abstract*The e!ect of refractive index on particle size distributions measured by optical particle counters is examined. Similar to previous investigations, it is found that optical counters undersize ambient particles because the refractive index of these particles is generally lower than that of the latex particles commonly used for the calibration of optical counters. The maximum undersizing is found to occur when particle sizes are comparable to the wavelength of light used in the measurement. A new approach for modeling the e!ect of refractive index on the sizing of optical counters is presented. Previously derived optical response functions are compared and a generalized formulation is proposed which includes the existing response functions as special cases. Algorithms are presented for correcting size distributions measured by optical counters for the di!erence between the refractive index of ambient and calibration particles. Data collected by a Passive Cavity Aerosol Spectrometer (PCASP) and by an integrating nephelometer are compared. Light scattering coe$cients calculated from the optical probe data uncorrected for the e!ect of refractive index di!er from those measured by the integrating nephelometer by a factor of 2. An iterative procedure that adjusts the PCASP-measured size distribution for the e!ect of refractive index is used to derive the best agreement between calculated and observed light scattering coe$cients. The refractive indices of aerosols at wavelength of 0.45 km that best "t the data vary between 1.3 and 1.5, with an average of 1.41. The relative importance of the underestimation of light scattering coe$cients calculated from the PCASP-measured size distributions due to the refractive index and the size truncation e!ect are evaluated. The former is found to be more important than the latter. Implications of this study for addressing aerosol shortwave radiative forcing and potential uncertainties relevant to this study are discussed. Published by Elsevier Science Ltd
1 . I NT RO D UC T IO N
Light scattering coe$cients can be measured directly with an integrating nephelometer (Heintzenberg and Charlson, 1996). They can also be calculated by applying Mie theory to known aerosol size distributions measured by an optical particle counter (OPC) such as the Passive Cavity Aerosol Spectrometer (PCASP) (Particle Measuring System, Inc., Boulder, CO). A number of studies have compared the two methods in the so-called closure experiments and good correlations have been reported (Ensor et al., 1972; Hegg et al., 1996; Anderson et al., 1996; Quinn et al., 1996). However, light scattering coe$cients calculated from size distributions measured by OPCs have been found to be systematically lower than those measured with nephelometers. For this reason the credibility of OPCs for calculation of light scattering coe$cients has been questioned (Wilson et al., 1988; Eldering et al., 1994; Hegg et al., 1996). OPCs are often calibrated with latex particles having a refractive index (m) of 1.588. Because refractive indices of real aerosols are often less, diameters measured by OPCs will be smaller than the real ones. This underestimation of particle size will in turn cause the calculated light scattering coe$cients to be low. Recent studies using ambient particles shows that such undersizing is a function of particle size and reaches a maximum in the
* By acceptance of this article, the publisher and/or recipient acknowledges the U.S. government's right to retain a non-exclusive, royalty-free license in and to any copyright covering this paper. This research was performed under the auspices of the U.S. Department of Energy under Contract No. DE-AC02-98CH10886. R Author to whom correspondence should be addressed. 945
946
Y. Liu and P.H. Daum
region where particle sizes are comparable to the wavelength of the light (e.g., 0.628 km) used for the measurement (Covert et al., 1990; Hering and McMurry, 1991; Stolzenburg et al., 1998). The occurrence of maximum undersizing in this region should be emphasized because particles of this diameter are also the most e$cient at scattering shortwave solar radiation (Waggoner et al., 1981; Schwartz, 1996). The "rst objective of this paper is to develop a theoretical approach for correcting particle size distributions measured by OPCs for the e!ect of the di!erence between the refractive index of particles used in the calibration of the counter and that of ambient aerosols, with focus on the PCASP. Our second objective is to show that light scattering coe$cients calculated by applying Mie theory to PCASP-measured size distributions can be reconciled with the nephelometer measurements by taking into account the di!erence between the refractive index of the calibration and ambient particles. Furthermore, we demonstrate that refractive indices of atmospheric aerosols can be estimated by matching the nephelometer measured-light scattering coe$cients with those calculated from the PCASP-measured size distributions if particle sizes are corrected for refractive index by use of our new approach. The approach is applied to the analysis of data collected during a recent Intensive Observation Period (IOP) over the Southern Great Plains (SGP) Atmospheric Radiation Measurement (ARM) site in the spring of 1997 (1997-Spring IOP hereafter). Our third objective is to compare the refractive index e!ect with the size truncation e!ect arising from the limited size range (nominally between 0.10 and 3.2 km diameter) over which the PCASP measures particles. This e!ect has been cited as a reason for the underestimation of light scattering coe$cients calculated from particle size distributions measured with an optical particle counter (Hegg et al., 1996). To the best of our knowledge, however, no studies have quantitatively compared the relative importance of these two sources of error. The paper is organized as follows. Theoretical models of the PCASP optical response are discussed in Section 2. The e!ect of refractive index is investigated, and a simple model is developed to correct the measured size distributions for the e!ect of the di!erence between the refractive index of ambient and calibration particles. A generalized response function is proposed to describe the optical response of the PCASP. Section 3 applies the model to a local closure experiment conducted during the 1997-Spring IOP. Refractive indices of ambient aerosols are estimated. In Section 4, the size truncation e!ect is compared with the refractive index e!ect. Major conclusions and implications of this study are outlined in Section 5.
2 . A NE W T HE O RET I CA L AP PR OA C H FO R S I ZE CO RRE CTI ON
2.1. ¹heoretical models for response functions A key to accounting for the e!ect of refractive index on the sizing of the PCASP is modeling its optical response which, for this instrument, is complicated by the use of standing wave technology. Two theoretical formulations have been proposed. Pinnick and Auvermann (1979) expressed the response function as
n F ["S (D, m, h)#S (D, m, n!h)" R (D, m)" k F #"S (D, m, h)#S (D, m, n!h)"] sin h dh,
(1)
where D represents the particle diameter; h is the scattering angle, with h "353 and h "1203 for the PCASP; i"2n/j is the wave number (j is the wavelength), and S ( . ) and S ( . ) are the complex scattering amplitude functions corresponding to light polarized with electric "eld perpendicular and parallel to the scattering plane respectively. Garvey and Pinnick (1983) proposed a slightly di!erent expression that adds the scattering intensities
The e!ect of refractive index on size distributions
947
instead of the scattering amplitudes:
n F R (D, m)" ["S (D, m, h)"#"S (D, m, n!h)" k F #"S (D, m, h)"#"S (D, m, n!h)"] sin h dh, (2) They designated R as the summed-amplitude response (hereafter SAR) and R as the summed-"eld response (hereafter SFR). The only comparison of these two response functions with experimental results was made by Garvey and Pinnick (1983), who concluded that `the experimental errors were su$ciently large that both SAR and SFR can be "t to the data equally wella. In this paper, instead of comparing the response functions directly, we compare the performance of SAR and SFR by comparing the ratio of the apparent diameter measured by an OPC to the real diameter (diameter ratio hereafter) as a function of apparent diameter so that the e!ects of unknown uncertainties associated with the instrumentation is minimized. It is well known that the response functions given above are multivalued. Usually a smoothing algorithm is used to overcome this problem (Kim and Boatman, 1990). Here we "t the response functions to polynomials over the range of refractive indices of atmospheric aerosols (m"1.588, 1.5, 1.45, 1.4, and 1.3), and "nd that a polynomial of order 8 gives a good "t: D" c RG, (3) G G where c is the "tting coe$cient. The diameter is expressed as a function of the response so G that the diameter correction can be conveniently made. With the polynomials for di!erent refractive indices as described by equation (3), the diameter ratio is readily calculated given the optical responses. Note that in our calculations throughout this paper the prefactor (n/i) is neglected because it does not a!ect the calculated diameter ratios.
2.2. E+ect of refractive index Tables 1 and 2 list the "tting coe$cients and correlation coe$cients at di!erent refractive indices for the SAR and SFR formulations, respectively. The diameter ratio calculated by use of these polynomials is shown in Figs 1 and 2 for SAR and SFR, respectively, as a function of apparent diameter. The values (crosses) of the diameter ratio for ambient aerosols reported by Stolzenburg et al. (1998) are also shown for comparison. It can be seen that the measured values fall within our model results over the range of refractive indices expected for ambient aerosols, and that a minimum in the diameter ratio occurs near the resonance size range. Stolzenburg et al. (1998) found an e!ective refractive index of
Table 1. Coe$cients of "tting polynomials for summed-amplitude response function M C C C C C C C C C RH
1.588 1.1128e!1 1.0249e!3 !1.2132e!6 8.1732e!10 !2.7645e!13 5.1425e!17 !5.3699e!21 2.9503e!25 !6.6316e!30 0.939
1.5 1.2627e!1 9.0409e!4 !7.8478e!7 4.7729e!10 !1.5836e!13 3.0351e!17 !3.3887e!21 2.0429e!25 !5.1179e!30 0.948
* R represents the correlation coe$cient of the "tting.
1.4 1.5660e!1 6.6678e!4 2.8696e!7 !5.6142e!10 2.7784e!13 !6.4849e!17 7.9082e!21 !4.8727e!25 1.1987e!29 0.981
1.3 1.4918e!1 1.6359e!3 !1.4638e!6 9.3605e!10 !3.8678e!13 9.9916e!17 !1.5316e!20 1.2631e!24 !4.3004e!29 0.995
948
Y. Liu and P.H. Daum Table 2. Coe$cients of "tting polynomials for summed-"eld response function m C C C C C C C C C RH
1.588 1.3813e!1 1.1847e!3 !2.0440e!6 2.2126e!9 !1.2806e!12 4.1431e!16 !7.4677e!20 6.9909e!24 !2.6467e!28 0.987
1.5 1.4510e!1 1.3033e!3 !2.0506e!6 2.0404e!9 !1.0829e!12 3.1879e!16 !5.2222e!20 4.4591e!24 !1.5490e!28 0.995
1.4 1.6042e!1 1.3802e!3 !1.7220e!6 1.5101e!9 !7.4151e!13 2.0463e!16 !3.1533e!20 2.5332e!24 !8.2689e!29 0.998
1.3 1.7193e!1 1.9822e!3 !2.9230e!6 3.2284e!9 !2.0825e!12 7.6699e!16 !1.5929e!19 1.7391e!23 !7.7732e!28 0.999
* R represents the correlation coe$cient of the "tting.
Fig. 1. Diameter ratio as a function of the apparent diameter measured with the PCASP when the SAF formulation for the optical response is used. The crosses are measurements from Stolzenburg et al. (1998).
1.45$0.03 for the ambient aerosol particles by matching the response of an optical counter to size-selected ambient particles under the assumption that absorption may be neglected and that the particle size does not change as a result of heating. It is interesting to note that our model results are consistent with their measurements of diameter ratios when m"1.45 is used. This suggests that refractive index may be estimated by matching diameter ratio measurements with our model results. A further examination of both "gures reveals that the SFR formulation models the overall trend better than the SAR formulation (SAR gives peaks at small particle sizes that are contrary to observations), whereas SAR-modeled diameter ratios are closer to the measured minimum near the resonance diameter region. That the SFR formulation represents the trend better than the SAR formulation is surprising because the latter is believed to be more physically based than the former for modeling the response of optical counters (Garvey and Pinnick, 1983). However, the
The e!ect of refractive index on size distributions
949
Fig. 2. Same as Fig. 1, except for the SFR formulation.
justi"cation for use of the SAR formulation is based on Mie theory as applied to an ideal instrumental con"guration which does not hold in practice. A real probe of this type cannot resolve the details of the resonance structure of the exact theoretical response, and smoothing is required to dampen the #uctuations of the response functions. Furthermore, large #uctuations in the SAR response function, make it more di$cult to smooth the response. This is indicated by the correlation coe$cients given in Tables 1 and 2: the correlation coe$cients for SAR are always lower than those for the SFR formulation. This point is also illustrated in Fig. 3. where it can be noted that while the SFR and SAR formulations give responses that exhibit the same general trend, the SFR response is much smoother. The better performance of SFR relative to SAR can also be argued from a physical perspective. Further examination of equations (1) and (2) reveals that SAR can be expressed as a sum of SFR and a term *R: R "R #*R,
(4a)
*R"2 [a (h)a(n!h)#b (h)b (n!h)#a (h)a (n!h) #b (h)b (n!h)] sin h dh, (4b) where a and b represent the real and imaginary part of the scattering amplitude function (Van de Hulst, 1980). The term *R actually describes the contribution to the response associated with light interference. As demonstrated in Fig. 4, this term is highly #uctuating and small in magnitude compared to SFR. The interference may exist for the ideal case; but it will be dampened (even eliminated) in practical instrumentation. Schuster and Knollenberg (1972) pointed out that the interferometric behavior of the laser cavity tends to suppress the normal Mie resonance. The other di$culty rests with the detection of phase di!erences in the scattering of standing-laser waves (Arnott, 1998, private communication). Although the results somehow support the use of the SFR formulation, the SAR formulation cannot be completely ruled out. A speci"c response may as well fall between
950
Y. Liu and P.H. Daum
Fig. 3. Optical responses as a function of particle diameter for both SAR (crosses) and SFR ("lled dots) formulation. Refractive index m"1.5 is used in the calculations.
Fig. 4. Comparison of the term *R (dotted curve) against the SFR response (solid curve), showing that the *R term #uctuates around zero, with absolute values much smaller than the SFR response.
The e!ect of refractive index on size distributions
951
the two extremes. The complexity of non-ideal conditions even prevents the manufacture (PMS) from providing a robust theoretical response function for the probe of this type (PMS PCASP-100X Operating Manual). Based on the relationship between the SAR and SFR formulation as described by equation (4), the optical response of a speci"c probe be expressed as R"R #a *R, (5) where a is an empirical `constanta ranging from 1 to 0. Evidently, this new formulation includes the existing expressions as extreme cases. The SAR formulation corresponds to a"1 and SFR to a"0. This generalized formulation may provide an approach for characterizing response of a speci"c probe by empirically determining the value of a. 3. ES TI M ATI ON O F TH E RE FR ACT IVE I N DE X AN D LO CAL CLO SU RE O F L I G HT SC A TT ERI NG CO E FF ICI E NTS
A number of studies have been made to compare nephelometer-measured light scattering coe$cients with those calculated from given size distributions. A common result of such studies is that calculated light scattering coe$cients are systematically lower than nephelometer measurements if size distributions measured by OPCs are used (e.g. Ensor et al., 1972; Hegg et al., 1996) whereas there is no such underestimation if scattering coe$cients are calculated from size distributions measured by non-optical counters (e.g. Quinn et al., 1995; Anderson et al., 1996). E!orts have been made to account for the e!ect of refractive index associated with OPCs. In a laboratory experiment, Wilson et al. (1988) compared light scattering coe$cients of aerosols of known composition measured with a nephelometer to those calculated from size distributions measured by an OPC and an electrical aerosol analyzer. They showed that after correcting the OPC data for refractive index, calculated light scattering coe$cients were closer to those measured by the nephelometer. Eldering et al. (1994) considered the refractive index e!ect according to the diameter ratio measurements made by Hering and McMurry (1991). Stolzenburg et al. (1998) measured the diameter ratio by combining an OPC with a di!erential mobility analyzer. They demonstrated that the agreement between light scattering coe$cients measured by a nephelometer and those calculated from the OPC data was improved when the OPC data was corrected for the refractive index e!ect by use of the diameter ratio measurements. Although these studies have demonstrated the importance of the refractive index e!ect, their usefulness is limited because diameter ratio measurements are often not available, and the refractive index can vary greatly with time and location. In this section we examine the nephelometer and PCASP data collected during 1997Spring IOP. The objective of this analysis is to demonstrate a new approach for accounting for the di!erence between the refractive index of the particles used to calibrate the PCASP probe, and ambient aerosols. The data were collected using the DOE Gulfstream-1 aircraft equipped with sensors for measuring temperature, pressure, humidity, solar radiation, and aerosol and cloud properties, including a PCASP and an integrating nephelometer. A Science Engineering Associates (SEA) Model 200 data acquisition system was used to collect data at a rate of 1 Hz. The aircraft #ew a total of 15 missions that covered a variety of weather conditions. Typically, the aircraft conducted a step pro"le from about 100 m above the surface to an altitude of approximately 5 km. Only four #ights in which the relative humidities were always less than 75% were analyzed to minimize the di!erent `dryinga e!ects of the integrating nephelometer and the PCASP. To alleviate the problem caused by di!erences in response times and sampling volumes between the nephelometer and the PCASP, we performed the calculations with 100 m-averaged data containing at least 50 data points. A total of 30 datasets from the four #ights satis"ed the criteria outlined above. The procedure that we used to analyze the data is as follows. First we adjusted the PCASP-measured size distributions for refractive indices m"1.3, 1.4, 1.45, 1.5, 1.588 (this represents the range of refractive indices expected for ambient aerosols). This is discussed in Section 2, and corrected diameters are summarized in Table 3. Then for each of these
952
Y. Liu and P.H. Daum Table 3. Corrected diameters (km) at di!erent refractive indicesH Apparent D
D (m"1.5)
D (m"1.45)
D (m"1.4)
D (m"1.3)
0.11 0.13 0.16 0.19 0.23 0.28 0.35 0.45 0.60 0.80 1.05 1.35
0.11 0.14 0.17 0.20 0.25 0.31 0.39 0.51 0.67 0.89 1.08 1.35
0.12 0.14 0.18 0.21 0.26 0.32 0.41 0.55 0.73 0.95 1.15 1.35
0.12 0.15 0.18 0.22 0.28 0.35 0.45 0.60 0.81 1.04 1.23 1.35
0.13 0.16 0.20 0.25 0.34 0.43 0.56 0.78 1.07 1.29 1.45 1.67
* Note: Corrected diameter equals the apparent diameter when D'1.35 km.
modi"ed size distributions and their associated refractive index, we calculated the light scattering coe$cients using
n DQ (D, m)nG(D) dD, pG " 4
(6)
where the superscript `i a denotes the ith measurement, the integration is taken over all the particles sampled by the PCASP, and the scattering e$ciency factor Q is calculated by use of Mie theory (Van de Hulst, 1980). The calculated light scattering coe$cients are then compared to those measured with the nephelometer and the refractive index best "tting the data is chosen. The agreement between p and that measured by the nephelometer (p ) is quanti"ed by the least-squares di!erence de"ned by , e " (pG !pG ), (7) G where N represents the total number of data used in the calculation. A similar idea has been used to estimate refractive indices of aerosol particles (Mathai and Harrison, 1980); however no size correction for refractive index was made in their study. In Table 4 we list the refractive indices which allow the closest agreement between the scattering coe$cient measured by the nephelometer and those calculated from the PCASP data, and the altitudes at which the data were collected. The refractive indices vary between 1.3 and 1.5, with an average of &1.41. There is no obvious height dependence of refractive indices. Note that instances where nephelometer-measured light scattering coe$cients are smaller than 1 M m\ are not considered here to minimize the e!ect of uncertainties associated with the nephelometer. This constrains our analysis mostly to boundary layer measurements. It is noteworthy that because the maximum refractive index e!ect is in the vicinity of the resonance diameter a refractive index smaller than that (1.588) of the latex particles used in the calibration of the PCASP is needed to increase the calculated light scattering coe$cients. This is demonstrated in Fig. 5, which shows the change with height of light scattering coe$cients measured by the nephelometer and those calculated with di!erent refractive indices for #ight 970415a. However, if the apparent PCASP-measured size distributions are not adjusted for refractive index, an increase in the refractive index is required to match measured and calculated scattering coe$cients, and the estimated refractive index will be larger than the refractive indices typically ascribed to ambient aerosol particles. Figure 6 shows the relationship of the light scattering coe$cients measured by the nephelometer with those calculated from the PCASP measured size distributions corrected by means of our procedure at the estimated refractive indices given in Table 4. Also shown are those calculated from the apparent PCASP-measured size distributions assuming
The e!ect of refractive index on size distributions
953
Table 4. Summary of estimated refractive indices 970413aH
970414bH
970415aH
970418aH
Height (m)
m
Height (m)
m
Height (m)
m
Height (m)
m
439 732 1044
1.45 1.45 1.45
444 732 1057 1353 1670 1971 2281
1.45 1.45 1.45 1.40 1.35 1.35 1.40
749 1069 1380 1686 1983 2014 2290 2314 2590 2616
1.40 1.40 1.45 1.45 1.45 1.50 1.45 1.45 1.40 1.40
482 529 787 811 1121 1429 1735 2045 2364 2659
1.30 1.40 1.30 1.30 1.30 1.40 1.40 1.40 1.40 1.45
* Note: 970413a, 970414b, 970415a, and 970418a denote #ight numbers.
Fig. 5. An example of vertical pro"le of calculated and measured light scattering coe$cients. Data are from the ascent of the #ight 970415a. The thick line shows the pro"le of the measured light scattering coe$cients; the remaining lines show those calculated from the PCASP measurements for various refractive indices. It is evident that the refractive index needs to be less than 1.588 for the calculated and measured light scattering coe$cients to match.
a refractive index of m"1.45. It is clear from this "gure that the namK ve use of the apparent PCASP measured-size distributions can cause as much as &50% negative error in calculation of the light scattering coe$cients. Our new approach for accounting for the e!ect of refractive index signi"cantly improves the closure of light scattering coe$cients. 4 . Q UA NT I F I CAT IO N OF SO U RCE S OF U N DE RE STI MAT IN G L IG H T SC ATT ER IN G C OE FF I CI EN TS CAL C ULAT ED FRO M PCA S P- ME ASU RE D SI Z E D IS T RIB U TIO N S
The PCASP sizes particles between 0.1 and 3.2 km and yet the nephelometer measures light scattered from particles outside this range (Heintzenberg and Charlson, 1996). The e!ect of the size truncation has been suggested as a reason for the underestimation of light
954
Y. Liu and P.H. Daum
Fig. 6. Closure experiments between nephelometer-measured light scattering coe$cients and those calculated from the PCASP-measured size distributions. The crosses (dots) represent the cases when the apparent (corrected) size distributions are used in the calculation of light scattering coe$cients. Also shown are the "tting equations, where p is the measured light scattering coe$cient, and
p and p are values calculated from corrected and uncorrected PCASP measured size distributions.
scattering coe$cients calculated from PCASP-measured size distributions by Hegg et al. (1996). To the best of our knowledge, no studies have been made to quantitatively estimate the importance of this e!ect relative to the refractive index e!ect for typical ambient aerosol size distributions. This section serves to "ll this gap. An evaluation of the relative importance of the two e!ects requires robust knowledge of diameter ratio, refractive index and particle concentration between 0.05 and 0.1 km. Since these pieces of information are not at our disposal, we examine these e!ects using an approximate method. We "rst obtain a size distribution by averaging all the size distributions of aerosols with estimated refractive index of m"1.45 directly measured by the PCASP (without size correction), and then "t this distribution with a power-law function as shown by Fig. 7. The best "t power-law distribution (with a correlation coe$cient of 0.96) is n(D)"0.8646D\ .
(8)
This power-law distribution is used in subsequent calculations. The apparent and the corrected light scattering coe$cient are calculated from the apparent and corrected diameter of the PCASP respectively. The contribution due to the refractive index e!ect is the di!erence between the two. The contribution from small particles with diameters between 0.05 and 0.1 km was calculated by extrapolating equation (8) to 0.05 km and repeating the calculation. The contributions due to the refractive index e!ect and the size truncation e!ect relative to the apparent light scattering coe$cient calculated by use of the PCASP apparent diameter are &60% and &2%, respectively. Therefore, the refractive index e!ect dominates the underestimation of calculated light scattering coe$cients. It should be noted that these estimates accurately holds only for aerosols as described by equation (8); deviations are anticipated for real aerosols with di!erent size distributions and refractive indices. Nevertheless, the basic conclusion is believed to hold.
The e!ect of refractive index on size distributions
955
Fig. 7. The size distribution obtained by averaging all the cases with m"1.45. The "tting power law is also given.
Fig. 8. Light scattering coe$cient normalized to the maximum value as a function of particle diameter at the three wavelengths of the nephelometer. The size distribution described by equation (8) and m"1.45 are used in the calculation. A combination of this "gure with Fig. 2 demonstrates that the maximum undersizing of the PCASP corresponds to the size region where maximum light scattering occurs.
The above results can be further understood by examining Fig. 8 together with Fig. 2. Figure 8 displays the distribution (at the three wavelengths of the nephelometer) of light scattering coe$cient normalized to the corresponding maximum values with respect to
956
Y. Liu and P.H. Daum
particle diameter, showing that the contribution from particles less than 0.1 km is very small. Furthermore, a combination of Figs 8 and 2 shows that the diameter range of the maximum refractive index e!ect corresponds to that over which particles scatter visible light most e$ciently. 5. C ON CL U DI NG RE MA RK S
Two theoretical functions (SAR and SFR) for representing the optical response of the PCASP are compared in terms of diameter ratio as a function of apparent diameter. The SFR formulation is found to be somewhat better than the more frequently used SAR formulation. It is argued that the SFR formulation is better, plausibly because it does not attempt to represent the resonance structure that exists for ideal Mie scattering. There is good evidence to believe that this resonance structure is dampened by scattering of standing waves in an optical particle counter, and that these phase di!erences are not present in practical instrumentation. A generalized response function is proposed which includes the existing functions as special cases. A new approach for deriving the refractive index of ambient aerosols and correcting the PCASP-measured size distributions is developed. The results from this approach are in agreement with the limited ambient measurements of diameter ratios that have been made, and clearly demonstrate the existence of a minimum in the diameter ratio near the resonance diameter region. The new approach is used to improve the closure experiment between light scattering coe$cients calculated by applying Mie theory to aerosol size distributions measured with the PCASP and those directly measured with an integrating nephelometer. It is found that the use of uncorrected PCASP-measured size distributions causes calculated light scattering coe$cients to be signi"cantly underestimated. Correcting PCASP-measured size distributions for the di!erence between the refractive index of calibration and ambient aerosols yields light scattering coe$cients that agree quite well with those measured by the integrating nephelometer. Refractive indices of ambient particles are estimated using this procedure and are found to vary between 1.3 and 1.5, with an average of 1.41. Such values are within the range of refractive indices measured, or estimated for ambient aerosols by other means. The relative magnitude of the refractive index, and size truncation e!ects on the PCASPderived light scattering coe$cients are compared. It is found that the refractive index e!ect predominates. This study shows that undersizing of particles by optical counters can lead to signi"cant negative errors in calculation of ambient light scattering because optical counters undersize aerosol particles mostly in the size range where particles are most e$cient at scattering solar radiation. These "ndings could be important for the assessment of the e!ects of aerosols on climate change, because data from the PCASP and/or its equivalent are widely used to estimate the e!ect of aerosols on the amount of solar radiation reaching the Earth's surface. It should be noted that neither aerosol absorption nor particle non-sphericity has been considered in this analyis. Although modeling the e!ect of absorption on the diameter ratio is straightforward, estimating imaginary parts of refractive indices requires additional data that are not available. We speculate that the e!ect of absorption is somewhat equivalent to lowering the real part of refractive index. Therefore, the estimated refractive indices reported in this paper may be somewhat lower than the actual values, depending on the degree of absorption of the aerosols. The e!ect of non-sphericity of aerosol particles on light scattering and subsequent results introduces further complication. Although the nonsphericity e!ect on total light scattering may be small, the phase function is rather sensitive to particle shapes (Mishchenko et al., 1997). Therefore, particle nonsphericity may a!ect angular truncation e!ect of integrating nephelometers. Particle nonsphericity may also a!ect the performance of the PCASP, among others, by reducing the resonance structure of light scattering (Liu et al., 1998,1999). Unfortunately, the issue of particle non-sphericity is almost intractable at present due to the lack of information on particle shapes and the di$culty involved in calculating light scattering by non-spherical particles. The counting
The e!ect of refractive index on size distributions
957
e$ciency of an OPC, and the mixing nature of ambient aerosols may also cause errors in size distribution measurements and calculated light scattering coe$cients. Acknowledgements*We are indebted to Dr. L. Newman at Brookhaven National Laboratory, who have carefully read the manuscript and provided valuable suggestions. We would like to acknowledge constructive discussions with Drs. S. E. Schwartz, R. McGraw, R. N. Halthore, Y.-N. Lee, D. Imre, F. Brechtel, and S. R. Springston at the Brookhaven National Laboratory. Comments by the anonymous referees were very constructive and insightful. This research was supported by the Environmental Sciences Division of the U.S. Department of Energy, as part of the Atmospheric Radiation Measurements Program and was performed under contract DE-AC03-98CH10886.
REF ER E NCE S Anderson, T. L., Covert, D. S. and Marshall, S. F. et al. (1996) Performance characteristics of a high-sensitivity, three-wavelength, total scatter/back-scatter nephelometer. J. Atmos. Ocean. Technol. 13, 967}986. Arnott, W. P. (1998) Private communication. Covert, D. S., Heintzenberg, J. and Hansson, H. (1990) Electro-optical detection of external mixtures in aerosols. Aerosol Sci. Technol. 12, 446}456. Eldering, A., Cass, G. R. and Moon, K. C. (1994) An air monitoring network using continuous particle size distribution monitors: connecting pollutant properties to visibility via Mie scattering calculations. Atmos. Environ. 28, 2733}2749. Ensor, D. S., Charlson, R. J., Ahlquist, N. C., Whitby, K. T., Husar, R. B. and Liu, B. Y. H. (1972) Multiwavelength nephelometer measurements in Los Angeles smog aerosols: I. Comparison of calculated and measured lightscattering. J. Colloid Interface Sci. 39, 242}251. Garvey, D. M. and Pinnick, R. G. (1983) Response of characteristics of the Particle Measuring Systems Active Scattering Aerosol Spectrometeor Probe (ASASP-X). Aerosol Sci. Technol. 2, 477}488. Hegg, D. A., Hobbs, P. V., Gasso, S., Nance, J. D. and Rango, A. L. (1996) Aerosol measurements of some in the Arctic relevant to direct and indirect radiative forcing. J. Geophys. Res. D101, 23,349}23,363. Heintzenberg, J. and Charlson, R. J. (1996) Design and applications of the integrating nephelometer: a review. J. Atmos. Ocean. Technol. 13, 987}1000. Hering, S. V. and McMurry, P. H. (1991) Response of a PMS LAS-X laser optical particle counter to monodisperse atmospheric aerosols. Atmos. Environ. 25A, 463}468. Kim, Y. J. and Boatman, J. F. (1990) Size calibration corrections for the Active Scattering Aerosol Spectrometer Probe (ASASP-100X). Aerosol Sci. Technol. 12, 665}672. Liu, Y., Arnott, W. P. and Hallett, J. (1998) Anomalous di!raction theory for arbitrarily oriented "nite circular cylinders and comparison with exact T-matrix results. Appl. Opt. 37, 5019}5030. Liu, Y., Arnott, W. P. and Hallett, J. (1999) Particle size distribution retrieval from multispectral optical depth: In#uences of particle nonsphericity and refractive index. J. Geophys. Res. D104, 31753}31762. Mathai, C. V. and Harrison, A. W. (1980) Estimation of atmospheric aerosol refractive index. Atmos. Environ. 14, 1131}1135. Mishchenko, M. I., Travis, L. D., Kahn, R. A. and West, R. A. (1997) Modeling phase functions for dustlike tropospheric aerosols using a shape mixture of randomly oriented polydisperse spheroids. J. Geophys. Res. D102, 16,831}16,847. Pinnick, R. G. and Auvermann, H. J. (1979) Response characteristics of Knollenberg light-scattering aerosol counters. J. Aerosol Sci. 10, 55}74. Quinn, P. K., Marshall, S. F., Bates, T. S., Covert, D. S. and Kapusin, V. N. (1995) Comparison of measured and calculated aerosol properties relevant to the direct radiative forcing of tropospheric sulfate aerosol on climate. J. Gephys. Res. D100, 8977}8991. Quinn, P. K. et al. (1996) Closure in tropospheric aerosol-climate research: a review and future needs for addressing aerosol direct shortwave radiative forcing. Beitr. Phys. Atmos. 69, 547}577. Schwartz, S. E. (1996) The whitehouse e!ect*shortwave radiative forcing of climate by anthropogenic aerosols: an overview. J. Aerosol Sci. 27, 359}382. Schuster, B. G. and Knollenberg, R. (1972) Detection and sizing of small particles in an open cavity gas laser. App. Opt. 11, 1515}1520. Stolzenburg, M., Kreisberg, N. and Hering, S. (1998) Atmospheric size distributions measured by di!erential mobility optical particle size spectrometry. Aerosol Sci. Technol. 29, 402}418. Van de Hulst, H. C. (1980) Light Scattering by Small Particles, 470pp. Dover Publications, New York. Waggoner, A. P., Weiss, R. E., Ahlquist, N. C., Covert, D. S., Will, S. and Charlson, R. J. (1981) Optical characteristics of atmospheric aerosols. Atmos. Environ. 15, 1891}1909. Wilson, J. C., Gupta, A., Whitby, K. T. and Wilson, W. E. (1988) Measured aerosol light scattering coe$cients compared with values calculated from EAA and optical particle counter measurements: improving the utility of the comparison. Atmos. Environ. 22, 789}793.
J. Aerosol Sci. Vol. 31, No. 8, pp. 945}957, 2000 Published by Elsevier Science Ltd Printed in Great Britain 0021-8502/00/$ - see front matter
THE EFFECT OF REFRACTIVE INDEX ON SIZE DISTRIBUTIONS AND LIGHT SCATTERING COEFFICIENTS DERIVED FROM OPTICAL PARTICLE COUNTERS* Yangang LiuR and Peter H. Daum Brookhaven National Laboratory, Environmental Chemistry Division, Bldg. 815E, P.O. Box 5000, Upton, NY 11973-5000, USA (First received 7 June 1999; and in ,nal form 18 November 1999) Abstract*The e!ect of refractive index on particle size distributions measured by optical particle counters is examined. Similar to previous investigations, it is found that optical counters undersize ambient particles because the refractive index of these particles is generally lower than that of the latex particles commonly used for the calibration of optical counters. The maximum undersizing is found to occur when particle sizes are comparable to the wavelength of light used in the measurement. A new approach for modeling the e!ect of refractive index on the sizing of optical counters is presented. Previously derived optical response functions are compared and a generalized formulation is proposed which includes the existing response functions as special cases. Algorithms are presented for correcting size distributions measured by optical counters for the di!erence between the refractive index of ambient and calibration particles. Data collected by a Passive Cavity Aerosol Spectrometer (PCASP) and by an integrating nephelometer are compared. Light scattering coe$cients calculated from the optical probe data uncorrected for the e!ect of refractive index di!er from those measured by the integrating nephelometer by a factor of 2. An iterative procedure that adjusts the PCASP-measured size distribution for the e!ect of refractive index is used to derive the best agreement between calculated and observed light scattering coe$cients. The refractive indices of aerosols at wavelength of 0.45 km that best "t the data vary between 1.3 and 1.5, with an average of 1.41. The relative importance of the underestimation of light scattering coe$cients calculated from the PCASP-measured size distributions due to the refractive index and the size truncation e!ect are evaluated. The former is found to be more important than the latter. Implications of this study for addressing aerosol shortwave radiative forcing and potential uncertainties relevant to this study are discussed. Published by Elsevier Science Ltd
1 . I NT RO D UC T IO N
Light scattering coe$cients can be measured directly with an integrating nephelometer (Heintzenberg and Charlson, 1996). They can also be calculated by applying Mie theory to known aerosol size distributions measured by an optical particle counter (OPC) such as the Passive Cavity Aerosol Spectrometer (PCASP) (Particle Measuring System, Inc., Boulder, CO). A number of studies have compared the two methods in the so-called closure experiments and good correlations have been reported (Ensor et al., 1972; Hegg et al., 1996; Anderson et al., 1996; Quinn et al., 1996). However, light scattering coe$cients calculated from size distributions measured by OPCs have been found to be systematically lower than those measured with nephelometers. For this reason the credibility of OPCs for calculation of light scattering coe$cients has been questioned (Wilson et al., 1988; Eldering et al., 1994; Hegg et al., 1996). OPCs are often calibrated with latex particles having a refractive index (m) of 1.588. Because refractive indices of real aerosols are often less, diameters measured by OPCs will be smaller than the real ones. This underestimation of particle size will in turn cause the calculated light scattering coe$cients to be low. Recent studies using ambient particles shows that such undersizing is a function of particle size and reaches a maximum in the
* By acceptance of this article, the publisher and/or recipient acknowledges the U.S. government's right to retain a non-exclusive, royalty-free license in and to any copyright covering this paper. This research was performed under the auspices of the U.S. Department of Energy under Contract No. DE-AC02-98CH10886. R Author to whom correspondence should be addressed. 945
946
Y. Liu and P.H. Daum
region where particle sizes are comparable to the wavelength of the light (e.g., 0.628 km) used for the measurement (Covert et al., 1990; Hering and McMurry, 1991; Stolzenburg et al., 1998). The occurrence of maximum undersizing in this region should be emphasized because particles of this diameter are also the most e$cient at scattering shortwave solar radiation (Waggoner et al., 1981; Schwartz, 1996). The "rst objective of this paper is to develop a theoretical approach for correcting particle size distributions measured by OPCs for the e!ect of the di!erence between the refractive index of particles used in the calibration of the counter and that of ambient aerosols, with focus on the PCASP. Our second objective is to show that light scattering coe$cients calculated by applying Mie theory to PCASP-measured size distributions can be reconciled with the nephelometer measurements by taking into account the di!erence between the refractive index of the calibration and ambient particles. Furthermore, we demonstrate that refractive indices of atmospheric aerosols can be estimated by matching the nephelometer measured-light scattering coe$cients with those calculated from the PCASP-measured size distributions if particle sizes are corrected for refractive index by use of our new approach. The approach is applied to the analysis of data collected during a recent Intensive Observation Period (IOP) over the Southern Great Plains (SGP) Atmospheric Radiation Measurement (ARM) site in the spring of 1997 (1997-Spring IOP hereafter). Our third objective is to compare the refractive index e!ect with the size truncation e!ect arising from the limited size range (nominally between 0.10 and 3.2 km diameter) over which the PCASP measures particles. This e!ect has been cited as a reason for the underestimation of light scattering coe$cients calculated from particle size distributions measured with an optical particle counter (Hegg et al., 1996). To the best of our knowledge, however, no studies have quantitatively compared the relative importance of these two sources of error. The paper is organized as follows. Theoretical models of the PCASP optical response are discussed in Section 2. The e!ect of refractive index is investigated, and a simple model is developed to correct the measured size distributions for the e!ect of the di!erence between the refractive index of ambient and calibration particles. A generalized response function is proposed to describe the optical response of the PCASP. Section 3 applies the model to a local closure experiment conducted during the 1997-Spring IOP. Refractive indices of ambient aerosols are estimated. In Section 4, the size truncation e!ect is compared with the refractive index e!ect. Major conclusions and implications of this study are outlined in Section 5.
2 . A NE W T HE O RET I CA L AP PR OA C H FO R S I ZE CO RRE CTI ON
2.1. ¹heoretical models for response functions A key to accounting for the e!ect of refractive index on the sizing of the PCASP is modeling its optical response which, for this instrument, is complicated by the use of standing wave technology. Two theoretical formulations have been proposed. Pinnick and Auvermann (1979) expressed the response function as
n F ["S (D, m, h)#S (D, m, n!h)" R (D, m)" k F #"S (D, m, h)#S (D, m, n!h)"] sin h dh,
(1)
where D represents the particle diameter; h is the scattering angle, with h "353 and h "1203 for the PCASP; i"2n/j is the wave number (j is the wavelength), and S ( . ) and S ( . ) are the complex scattering amplitude functions corresponding to light polarized with electric "eld perpendicular and parallel to the scattering plane respectively. Garvey and Pinnick (1983) proposed a slightly di!erent expression that adds the scattering intensities
The e!ect of refractive index on size distributions
947
instead of the scattering amplitudes:
n F R (D, m)" ["S (D, m, h)"#"S (D, m, n!h)" k F #"S (D, m, h)"#"S (D, m, n!h)"] sin h dh, (2) They designated R as the summed-amplitude response (hereafter SAR) and R as the summed-"eld response (hereafter SFR). The only comparison of these two response functions with experimental results was made by Garvey and Pinnick (1983), who concluded that `the experimental errors were su$ciently large that both SAR and SFR can be "t to the data equally wella. In this paper, instead of comparing the response functions directly, we compare the performance of SAR and SFR by comparing the ratio of the apparent diameter measured by an OPC to the real diameter (diameter ratio hereafter) as a function of apparent diameter so that the e!ects of unknown uncertainties associated with the instrumentation is minimized. It is well known that the response functions given above are multivalued. Usually a smoothing algorithm is used to overcome this problem (Kim and Boatman, 1990). Here we "t the response functions to polynomials over the range of refractive indices of atmospheric aerosols (m"1.588, 1.5, 1.45, 1.4, and 1.3), and "nd that a polynomial of order 8 gives a good "t: D" c RG, (3) G G where c is the "tting coe$cient. The diameter is expressed as a function of the response so G that the diameter correction can be conveniently made. With the polynomials for di!erent refractive indices as described by equation (3), the diameter ratio is readily calculated given the optical responses. Note that in our calculations throughout this paper the prefactor (n/i) is neglected because it does not a!ect the calculated diameter ratios.
2.2. E+ect of refractive index Tables 1 and 2 list the "tting coe$cients and correlation coe$cients at di!erent refractive indices for the SAR and SFR formulations, respectively. The diameter ratio calculated by use of these polynomials is shown in Figs 1 and 2 for SAR and SFR, respectively, as a function of apparent diameter. The values (crosses) of the diameter ratio for ambient aerosols reported by Stolzenburg et al. (1998) are also shown for comparison. It can be seen that the measured values fall within our model results over the range of refractive indices expected for ambient aerosols, and that a minimum in the diameter ratio occurs near the resonance size range. Stolzenburg et al. (1998) found an e!ective refractive index of
Table 1. Coe$cients of "tting polynomials for summed-amplitude response function M C C C C C C C C C RH
1.588 1.1128e!1 1.0249e!3 !1.2132e!6 8.1732e!10 !2.7645e!13 5.1425e!17 !5.3699e!21 2.9503e!25 !6.6316e!30 0.939
1.5 1.2627e!1 9.0409e!4 !7.8478e!7 4.7729e!10 !1.5836e!13 3.0351e!17 !3.3887e!21 2.0429e!25 !5.1179e!30 0.948
* R represents the correlation coe$cient of the "tting.
1.4 1.5660e!1 6.6678e!4 2.8696e!7 !5.6142e!10 2.7784e!13 !6.4849e!17 7.9082e!21 !4.8727e!25 1.1987e!29 0.981
1.3 1.4918e!1 1.6359e!3 !1.4638e!6 9.3605e!10 !3.8678e!13 9.9916e!17 !1.5316e!20 1.2631e!24 !4.3004e!29 0.995
948
Y. Liu and P.H. Daum Table 2. Coe$cients of "tting polynomials for summed-"eld response function m C C C C C C C C C RH
1.588 1.3813e!1 1.1847e!3 !2.0440e!6 2.2126e!9 !1.2806e!12 4.1431e!16 !7.4677e!20 6.9909e!24 !2.6467e!28 0.987
1.5 1.4510e!1 1.3033e!3 !2.0506e!6 2.0404e!9 !1.0829e!12 3.1879e!16 !5.2222e!20 4.4591e!24 !1.5490e!28 0.995
1.4 1.6042e!1 1.3802e!3 !1.7220e!6 1.5101e!9 !7.4151e!13 2.0463e!16 !3.1533e!20 2.5332e!24 !8.2689e!29 0.998
1.3 1.7193e!1 1.9822e!3 !2.9230e!6 3.2284e!9 !2.0825e!12 7.6699e!16 !1.5929e!19 1.7391e!23 !7.7732e!28 0.999
* R represents the correlation coe$cient of the "tting.
Fig. 1. Diameter ratio as a function of the apparent diameter measured with the PCASP when the SAF formulation for the optical response is used. The crosses are measurements from Stolzenburg et al. (1998).
1.45$0.03 for the ambient aerosol particles by matching the response of an optical counter to size-selected ambient particles under the assumption that absorption may be neglected and that the particle size does not change as a result of heating. It is interesting to note that our model results are consistent with their measurements of diameter ratios when m"1.45 is used. This suggests that refractive index may be estimated by matching diameter ratio measurements with our model results. A further examination of both "gures reveals that the SFR formulation models the overall trend better than the SAR formulation (SAR gives peaks at small particle sizes that are contrary to observations), whereas SAR-modeled diameter ratios are closer to the measured minimum near the resonance diameter region. That the SFR formulation represents the trend better than the SAR formulation is surprising because the latter is believed to be more physically based than the former for modeling the response of optical counters (Garvey and Pinnick, 1983). However, the
The e!ect of refractive index on size distributions
949
Fig. 2. Same as Fig. 1, except for the SFR formulation.
justi"cation for use of the SAR formulation is based on Mie theory as applied to an ideal instrumental con"guration which does not hold in practice. A real probe of this type cannot resolve the details of the resonance structure of the exact theoretical response, and smoothing is required to dampen the #uctuations of the response functions. Furthermore, large #uctuations in the SAR response function, make it more di$cult to smooth the response. This is indicated by the correlation coe$cients given in Tables 1 and 2: the correlation coe$cients for SAR are always lower than those for the SFR formulation. This point is also illustrated in Fig. 3. where it can be noted that while the SFR and SAR formulations give responses that exhibit the same general trend, the SFR response is much smoother. The better performance of SFR relative to SAR can also be argued from a physical perspective. Further examination of equations (1) and (2) reveals that SAR can be expressed as a sum of SFR and a term *R: R "R #*R,
(4a)
*R"2 [a (h)a(n!h)#b (h)b (n!h)#a (h)a (n!h) #b (h)b (n!h)] sin h dh, (4b) where a and b represent the real and imaginary part of the scattering amplitude function (Van de Hulst, 1980). The term *R actually describes the contribution to the response associated with light interference. As demonstrated in Fig. 4, this term is highly #uctuating and small in magnitude compared to SFR. The interference may exist for the ideal case; but it will be dampened (even eliminated) in practical instrumentation. Schuster and Knollenberg (1972) pointed out that the interferometric behavior of the laser cavity tends to suppress the normal Mie resonance. The other di$culty rests with the detection of phase di!erences in the scattering of standing-laser waves (Arnott, 1998, private communication). Although the results somehow support the use of the SFR formulation, the SAR formulation cannot be completely ruled out. A speci"c response may as well fall between
950
Y. Liu and P.H. Daum
Fig. 3. Optical responses as a function of particle diameter for both SAR (crosses) and SFR ("lled dots) formulation. Refractive index m"1.5 is used in the calculations.
Fig. 4. Comparison of the term *R (dotted curve) against the SFR response (solid curve), showing that the *R term #uctuates around zero, with absolute values much smaller than the SFR response.
The e!ect of refractive index on size distributions
951
the two extremes. The complexity of non-ideal conditions even prevents the manufacture (PMS) from providing a robust theoretical response function for the probe of this type (PMS PCASP-100X Operating Manual). Based on the relationship between the SAR and SFR formulation as described by equation (4), the optical response of a speci"c probe be expressed as R"R #a *R, (5) where a is an empirical `constanta ranging from 1 to 0. Evidently, this new formulation includes the existing expressions as extreme cases. The SAR formulation corresponds to a"1 and SFR to a"0. This generalized formulation may provide an approach for characterizing response of a speci"c probe by empirically determining the value of a. 3. ES TI M ATI ON O F TH E RE FR ACT IVE I N DE X AN D LO CAL CLO SU RE O F L I G HT SC A TT ERI NG CO E FF ICI E NTS
A number of studies have been made to compare nephelometer-measured light scattering coe$cients with those calculated from given size distributions. A common result of such studies is that calculated light scattering coe$cients are systematically lower than nephelometer measurements if size distributions measured by OPCs are used (e.g. Ensor et al., 1972; Hegg et al., 1996) whereas there is no such underestimation if scattering coe$cients are calculated from size distributions measured by non-optical counters (e.g. Quinn et al., 1995; Anderson et al., 1996). E!orts have been made to account for the e!ect of refractive index associated with OPCs. In a laboratory experiment, Wilson et al. (1988) compared light scattering coe$cients of aerosols of known composition measured with a nephelometer to those calculated from size distributions measured by an OPC and an electrical aerosol analyzer. They showed that after correcting the OPC data for refractive index, calculated light scattering coe$cients were closer to those measured by the nephelometer. Eldering et al. (1994) considered the refractive index e!ect according to the diameter ratio measurements made by Hering and McMurry (1991). Stolzenburg et al. (1998) measured the diameter ratio by combining an OPC with a di!erential mobility analyzer. They demonstrated that the agreement between light scattering coe$cients measured by a nephelometer and those calculated from the OPC data was improved when the OPC data was corrected for the refractive index e!ect by use of the diameter ratio measurements. Although these studies have demonstrated the importance of the refractive index e!ect, their usefulness is limited because diameter ratio measurements are often not available, and the refractive index can vary greatly with time and location. In this section we examine the nephelometer and PCASP data collected during 1997Spring IOP. The objective of this analysis is to demonstrate a new approach for accounting for the di!erence between the refractive index of the particles used to calibrate the PCASP probe, and ambient aerosols. The data were collected using the DOE Gulfstream-1 aircraft equipped with sensors for measuring temperature, pressure, humidity, solar radiation, and aerosol and cloud properties, including a PCASP and an integrating nephelometer. A Science Engineering Associates (SEA) Model 200 data acquisition system was used to collect data at a rate of 1 Hz. The aircraft #ew a total of 15 missions that covered a variety of weather conditions. Typically, the aircraft conducted a step pro"le from about 100 m above the surface to an altitude of approximately 5 km. Only four #ights in which the relative humidities were always less than 75% were analyzed to minimize the di!erent `dryinga e!ects of the integrating nephelometer and the PCASP. To alleviate the problem caused by di!erences in response times and sampling volumes between the nephelometer and the PCASP, we performed the calculations with 100 m-averaged data containing at least 50 data points. A total of 30 datasets from the four #ights satis"ed the criteria outlined above. The procedure that we used to analyze the data is as follows. First we adjusted the PCASP-measured size distributions for refractive indices m"1.3, 1.4, 1.45, 1.5, 1.588 (this represents the range of refractive indices expected for ambient aerosols). This is discussed in Section 2, and corrected diameters are summarized in Table 3. Then for each of these
952
Y. Liu and P.H. Daum Table 3. Corrected diameters (km) at di!erent refractive indicesH Apparent D
D (m"1.5)
D (m"1.45)
D (m"1.4)
D (m"1.3)
0.11 0.13 0.16 0.19 0.23 0.28 0.35 0.45 0.60 0.80 1.05 1.35
0.11 0.14 0.17 0.20 0.25 0.31 0.39 0.51 0.67 0.89 1.08 1.35
0.12 0.14 0.18 0.21 0.26 0.32 0.41 0.55 0.73 0.95 1.15 1.35
0.12 0.15 0.18 0.22 0.28 0.35 0.45 0.60 0.81 1.04 1.23 1.35
0.13 0.16 0.20 0.25 0.34 0.43 0.56 0.78 1.07 1.29 1.45 1.67
* Note: Corrected diameter equals the apparent diameter when D'1.35 km.
modi"ed size distributions and their associated refractive index, we calculated the light scattering coe$cients using
n DQ (D, m)nG(D) dD, pG " 4
(6)
where the superscript `i a denotes the ith measurement, the integration is taken over all the particles sampled by the PCASP, and the scattering e$ciency factor Q is calculated by use of Mie theory (Van de Hulst, 1980). The calculated light scattering coe$cients are then compared to those measured with the nephelometer and the refractive index best "tting the data is chosen. The agreement between p and that measured by the nephelometer (p ) is quanti"ed by the least-squares di!erence de"ned by , e " (pG !pG ), (7) G where N represents the total number of data used in the calculation. A similar idea has been used to estimate refractive indices of aerosol particles (Mathai and Harrison, 1980); however no size correction for refractive index was made in their study. In Table 4 we list the refractive indices which allow the closest agreement between the scattering coe$cient measured by the nephelometer and those calculated from the PCASP data, and the altitudes at which the data were collected. The refractive indices vary between 1.3 and 1.5, with an average of &1.41. There is no obvious height dependence of refractive indices. Note that instances where nephelometer-measured light scattering coe$cients are smaller than 1 M m\ are not considered here to minimize the e!ect of uncertainties associated with the nephelometer. This constrains our analysis mostly to boundary layer measurements. It is noteworthy that because the maximum refractive index e!ect is in the vicinity of the resonance diameter a refractive index smaller than that (1.588) of the latex particles used in the calibration of the PCASP is needed to increase the calculated light scattering coe$cients. This is demonstrated in Fig. 5, which shows the change with height of light scattering coe$cients measured by the nephelometer and those calculated with di!erent refractive indices for #ight 970415a. However, if the apparent PCASP-measured size distributions are not adjusted for refractive index, an increase in the refractive index is required to match measured and calculated scattering coe$cients, and the estimated refractive index will be larger than the refractive indices typically ascribed to ambient aerosol particles. Figure 6 shows the relationship of the light scattering coe$cients measured by the nephelometer with those calculated from the PCASP measured size distributions corrected by means of our procedure at the estimated refractive indices given in Table 4. Also shown are those calculated from the apparent PCASP-measured size distributions assuming
The e!ect of refractive index on size distributions
953
Table 4. Summary of estimated refractive indices 970413aH
970414bH
970415aH
970418aH
Height (m)
m
Height (m)
m
Height (m)
m
Height (m)
m
439 732 1044
1.45 1.45 1.45
444 732 1057 1353 1670 1971 2281
1.45 1.45 1.45 1.40 1.35 1.35 1.40
749 1069 1380 1686 1983 2014 2290 2314 2590 2616
1.40 1.40 1.45 1.45 1.45 1.50 1.45 1.45 1.40 1.40
482 529 787 811 1121 1429 1735 2045 2364 2659
1.30 1.40 1.30 1.30 1.30 1.40 1.40 1.40 1.40 1.45
* Note: 970413a, 970414b, 970415a, and 970418a denote #ight numbers.
Fig. 5. An example of vertical pro"le of calculated and measured light scattering coe$cients. Data are from the ascent of the #ight 970415a. The thick line shows the pro"le of the measured light scattering coe$cients; the remaining lines show those calculated from the PCASP measurements for various refractive indices. It is evident that the refractive index needs to be less than 1.588 for the calculated and measured light scattering coe$cients to match.
a refractive index of m"1.45. It is clear from this "gure that the namK ve use of the apparent PCASP measured-size distributions can cause as much as &50% negative error in calculation of the light scattering coe$cients. Our new approach for accounting for the e!ect of refractive index signi"cantly improves the closure of light scattering coe$cients. 4 . Q UA NT I F I CAT IO N OF SO U RCE S OF U N DE RE STI MAT IN G L IG H T SC ATT ER IN G C OE FF I CI EN TS CAL C ULAT ED FRO M PCA S P- ME ASU RE D SI Z E D IS T RIB U TIO N S
The PCASP sizes particles between 0.1 and 3.2 km and yet the nephelometer measures light scattered from particles outside this range (Heintzenberg and Charlson, 1996). The e!ect of the size truncation has been suggested as a reason for the underestimation of light
954
Y. Liu and P.H. Daum
Fig. 6. Closure experiments between nephelometer-measured light scattering coe$cients and those calculated from the PCASP-measured size distributions. The crosses (dots) represent the cases when the apparent (corrected) size distributions are used in the calculation of light scattering coe$cients. Also shown are the "tting equations, where p is the measured light scattering coe$cient, and
p and p are values calculated from corrected and uncorrected PCASP measured size distributions.
scattering coe$cients calculated from PCASP-measured size distributions by Hegg et al. (1996). To the best of our knowledge, no studies have been made to quantitatively estimate the importance of this e!ect relative to the refractive index e!ect for typical ambient aerosol size distributions. This section serves to "ll this gap. An evaluation of the relative importance of the two e!ects requires robust knowledge of diameter ratio, refractive index and particle concentration between 0.05 and 0.1 km. Since these pieces of information are not at our disposal, we examine these e!ects using an approximate method. We "rst obtain a size distribution by averaging all the size distributions of aerosols with estimated refractive index of m"1.45 directly measured by the PCASP (without size correction), and then "t this distribution with a power-law function as shown by Fig. 7. The best "t power-law distribution (with a correlation coe$cient of 0.96) is n(D)"0.8646D\ .
(8)
This power-law distribution is used in subsequent calculations. The apparent and the corrected light scattering coe$cient are calculated from the apparent and corrected diameter of the PCASP respectively. The contribution due to the refractive index e!ect is the di!erence between the two. The contribution from small particles with diameters between 0.05 and 0.1 km was calculated by extrapolating equation (8) to 0.05 km and repeating the calculation. The contributions due to the refractive index e!ect and the size truncation e!ect relative to the apparent light scattering coe$cient calculated by use of the PCASP apparent diameter are &60% and &2%, respectively. Therefore, the refractive index e!ect dominates the underestimation of calculated light scattering coe$cients. It should be noted that these estimates accurately holds only for aerosols as described by equation (8); deviations are anticipated for real aerosols with di!erent size distributions and refractive indices. Nevertheless, the basic conclusion is believed to hold.
The e!ect of refractive index on size distributions
955
Fig. 7. The size distribution obtained by averaging all the cases with m"1.45. The "tting power law is also given.
Fig. 8. Light scattering coe$cient normalized to the maximum value as a function of particle diameter at the three wavelengths of the nephelometer. The size distribution described by equation (8) and m"1.45 are used in the calculation. A combination of this "gure with Fig. 2 demonstrates that the maximum undersizing of the PCASP corresponds to the size region where maximum light scattering occurs.
The above results can be further understood by examining Fig. 8 together with Fig. 2. Figure 8 displays the distribution (at the three wavelengths of the nephelometer) of light scattering coe$cient normalized to the corresponding maximum values with respect to
956
Y. Liu and P.H. Daum
particle diameter, showing that the contribution from particles less than 0.1 km is very small. Furthermore, a combination of Figs 8 and 2 shows that the diameter range of the maximum refractive index e!ect corresponds to that over which particles scatter visible light most e$ciently. 5. C ON CL U DI NG RE MA RK S
Two theoretical functions (SAR and SFR) for representing the optical response of the PCASP are compared in terms of diameter ratio as a function of apparent diameter. The SFR formulation is found to be somewhat better than the more frequently used SAR formulation. It is argued that the SFR formulation is better, plausibly because it does not attempt to represent the resonance structure that exists for ideal Mie scattering. There is good evidence to believe that this resonance structure is dampened by scattering of standing waves in an optical particle counter, and that these phase di!erences are not present in practical instrumentation. A generalized response function is proposed which includes the existing functions as special cases. A new approach for deriving the refractive index of ambient aerosols and correcting the PCASP-measured size distributions is developed. The results from this approach are in agreement with the limited ambient measurements of diameter ratios that have been made, and clearly demonstrate the existence of a minimum in the diameter ratio near the resonance diameter region. The new approach is used to improve the closure experiment between light scattering coe$cients calculated by applying Mie theory to aerosol size distributions measured with the PCASP and those directly measured with an integrating nephelometer. It is found that the use of uncorrected PCASP-measured size distributions causes calculated light scattering coe$cients to be signi"cantly underestimated. Correcting PCASP-measured size distributions for the di!erence between the refractive index of calibration and ambient aerosols yields light scattering coe$cients that agree quite well with those measured by the integrating nephelometer. Refractive indices of ambient particles are estimated using this procedure and are found to vary between 1.3 and 1.5, with an average of 1.41. Such values are within the range of refractive indices measured, or estimated for ambient aerosols by other means. The relative magnitude of the refractive index, and size truncation e!ects on the PCASPderived light scattering coe$cients are compared. It is found that the refractive index e!ect predominates. This study shows that undersizing of particles by optical counters can lead to signi"cant negative errors in calculation of ambient light scattering because optical counters undersize aerosol particles mostly in the size range where particles are most e$cient at scattering solar radiation. These "ndings could be important for the assessment of the e!ects of aerosols on climate change, because data from the PCASP and/or its equivalent are widely used to estimate the e!ect of aerosols on the amount of solar radiation reaching the Earth's surface. It should be noted that neither aerosol absorption nor particle non-sphericity has been considered in this analyis. Although modeling the e!ect of absorption on the diameter ratio is straightforward, estimating imaginary parts of refractive indices requires additional data that are not available. We speculate that the e!ect of absorption is somewhat equivalent to lowering the real part of refractive index. Therefore, the estimated refractive indices reported in this paper may be somewhat lower than the actual values, depending on the degree of absorption of the aerosols. The e!ect of non-sphericity of aerosol particles on light scattering and subsequent results introduces further complication. Although the nonsphericity e!ect on total light scattering may be small, the phase function is rather sensitive to particle shapes (Mishchenko et al., 1997). Therefore, particle nonsphericity may a!ect angular truncation e!ect of integrating nephelometers. Particle nonsphericity may also a!ect the performance of the PCASP, among others, by reducing the resonance structure of light scattering (Liu et al., 1998,1999). Unfortunately, the issue of particle non-sphericity is almost intractable at present due to the lack of information on particle shapes and the di$culty involved in calculating light scattering by non-spherical particles. The counting
The e!ect of refractive index on size distributions
957
e$ciency of an OPC, and the mixing nature of ambient aerosols may also cause errors in size distribution measurements and calculated light scattering coe$cients. Acknowledgements*We are indebted to Dr. L. Newman at Brookhaven National Laboratory, who have carefully read the manuscript and provided valuable suggestions. We would like to acknowledge constructive discussions with Drs. S. E. Schwartz, R. McGraw, R. N. Halthore, Y.-N. Lee, D. Imre, F. Brechtel, and S. R. Springston at the Brookhaven National Laboratory. Comments by the anonymous referees were very constructive and insightful. This research was supported by the Environmental Sciences Division of the U.S. Department of Energy, as part of the Atmospheric Radiation Measurements Program and was performed under contract DE-AC03-98CH10886.
REF ER E NCE S Anderson, T. L., Covert, D. S. and Marshall, S. F. et al. (1996) Performance characteristics of a high-sensitivity, three-wavelength, total scatter/back-scatter nephelometer. J. Atmos. Ocean. Technol. 13, 967}986. Arnott, W. P. (1998) Private communication. Covert, D. S., Heintzenberg, J. and Hansson, H. (1990) Electro-optical detection of external mixtures in aerosols. Aerosol Sci. Technol. 12, 446}456. Eldering, A., Cass, G. R. and Moon, K. C. (1994) An air monitoring network using continuous particle size distribution monitors: connecting pollutant properties to visibility via Mie scattering calculations. Atmos. Environ. 28, 2733}2749. Ensor, D. S., Charlson, R. J., Ahlquist, N. C., Whitby, K. T., Husar, R. B. and Liu, B. Y. H. (1972) Multiwavelength nephelometer measurements in Los Angeles smog aerosols: I. Comparison of calculated and measured lightscattering. J. Colloid Interface Sci. 39, 242}251. Garvey, D. M. and Pinnick, R. G. (1983) Response of characteristics of the Particle Measuring Systems Active Scattering Aerosol Spectrometeor Probe (ASASP-X). Aerosol Sci. Technol. 2, 477}488. Hegg, D. A., Hobbs, P. V., Gasso, S., Nance, J. D. and Rango, A. L. (1996) Aerosol measurements of some in the Arctic relevant to direct and indirect radiative forcing. J. Geophys. Res. D101, 23,349}23,363. Heintzenberg, J. and Charlson, R. J. (1996) Design and applications of the integrating nephelometer: a review. J. Atmos. Ocean. Technol. 13, 987}1000. Hering, S. V. and McMurry, P. H. (1991) Response of a PMS LAS-X laser optical particle counter to monodisperse atmospheric aerosols. Atmos. Environ. 25A, 463}468. Kim, Y. J. and Boatman, J. F. (1990) Size calibration corrections for the Active Scattering Aerosol Spectrometer Probe (ASASP-100X). Aerosol Sci. Technol. 12, 665}672. Liu, Y., Arnott, W. P. and Hallett, J. (1998) Anomalous di!raction theory for arbitrarily oriented "nite circular cylinders and comparison with exact T-matrix results. Appl. Opt. 37, 5019}5030. Liu, Y., Arnott, W. P. and Hallett, J. (1999) Particle size distribution retrieval from multispectral optical depth: In#uences of particle nonsphericity and refractive index. J. Geophys. Res. D104, 31753}31762. Mathai, C. V. and Harrison, A. W. (1980) Estimation of atmospheric aerosol refractive index. Atmos. Environ. 14, 1131}1135. Mishchenko, M. I., Travis, L. D., Kahn, R. A. and West, R. A. (1997) Modeling phase functions for dustlike tropospheric aerosols using a shape mixture of randomly oriented polydisperse spheroids. J. Geophys. Res. D102, 16,831}16,847. Pinnick, R. G. and Auvermann, H. J. (1979) Response characteristics of Knollenberg light-scattering aerosol counters. J. Aerosol Sci. 10, 55}74. Quinn, P. K., Marshall, S. F., Bates, T. S., Covert, D. S. and Kapusin, V. N. (1995) Comparison of measured and calculated aerosol properties relevant to the direct radiative forcing of tropospheric sulfate aerosol on climate. J. Gephys. Res. D100, 8977}8991. Quinn, P. K. et al. (1996) Closure in tropospheric aerosol-climate research: a review and future needs for addressing aerosol direct shortwave radiative forcing. Beitr. Phys. Atmos. 69, 547}577. Schwartz, S. E. (1996) The whitehouse e!ect*shortwave radiative forcing of climate by anthropogenic aerosols: an overview. J. Aerosol Sci. 27, 359}382. Schuster, B. G. and Knollenberg, R. (1972) Detection and sizing of small particles in an open cavity gas laser. App. Opt. 11, 1515}1520. Stolzenburg, M., Kreisberg, N. and Hering, S. (1998) Atmospheric size distributions measured by di!erential mobility optical particle size spectrometry. Aerosol Sci. Technol. 29, 402}418. Van de Hulst, H. C. (1980) Light Scattering by Small Particles, 470pp. Dover Publications, New York. Waggoner, A. P., Weiss, R. E., Ahlquist, N. C., Covert, D. S., Will, S. and Charlson, R. J. (1981) Optical characteristics of atmospheric aerosols. Atmos. Environ. 15, 1891}1909. Wilson, J. C., Gupta, A., Whitby, K. T. and Wilson, W. E. (1988) Measured aerosol light scattering coe$cients compared with values calculated from EAA and optical particle counter measurements: improving the utility of the comparison. Atmos. Environ. 22, 789}793.